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21: 22.11 Fourier and Hyperbolic Series

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22.11.4 $$\mathrm{cd}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.4

Permalink:

http://dlmf.nist.gov/22.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22: 6.3 Graphics

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Figure 6.3.3: $|E_1\left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-4\le y\le 4$. … Magnify 3D Help

23: 9.4 Maclaurin Series

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9.4.1 $$\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Ai}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Source:

Olver (1997b, p. 54)

A&S Ref:

10.4.2 (with 10.4.4 and 10.4.5)

Permalink:

http://dlmf.nist.gov/9.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

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9.4.2 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

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9.4.3 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

A&S Ref:

10.4.3 (with 10.4.4 and 10.4.5)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

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9.4.4 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

24: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).

25: 22.10 Maclaurin Series

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22.10.1 $$\mathrm{sn}(z,k)=z-\left(1+k^2\right)\frac{z^3}{3!}+\left(1+14k^2+k^4\right)\frac{z^5}{5!}-\left(1+135k^2+135k^4+k^6\right)\frac{z^7}{7!}+O\left(z^9\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.1

Permalink:

http://dlmf.nist.gov/22.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

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22.10.2 $$\mathrm{cn}(z,k)=1-\frac{z^2}{2!}+\left(1+4k^2\right)\frac{z^4}{4!}-\left(1+44k^2+16k^4\right)\frac{z^6}{6!}+O\left(z^8\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.2

Permalink:

http://dlmf.nist.gov/22.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

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22.10.3 $$\mathrm{dn}(z,k)=1-k^2\frac{z^2}{2!}+k^2\left(4+k^2\right)\frac{z^4}{4!}-k^2\left(16+44k^2+k^4\right)\frac{z^6}{6!}+O\left(z^8\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $!$: factorial (as in $n!$), $z$: complex and $k$: modulus

A&S Ref:

16.22.3

Permalink:

http://dlmf.nist.gov/22.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(i), §22.10 and Ch.22

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22.10.4 $$\mathrm{sn}(z,k)=\sin z-\frac{k^2}{4}(z-\sin z\cos z)\cos z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.1

Referenced by:

§22.10(ii), §22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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22.10.5 $$\mathrm{cn}(z,k)=\cos z+\frac{k^2}{4}(z-\sin z\cos z)\sin z+O\left(k^4\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex and $k$: modulus

A&S Ref:

16.13.2

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.10(ii), §22.10 and Ch.22

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26: 4.43 Cubic Equations

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$A={\left(-\frac{4}{3}p\right)}^{1/2},$

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$B={\left(\frac{4}{3}p\right)}^{1/2}.$

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$A\sin a$, $A\sin \left(a+\frac{2}{3}\pi \right)$, and $A\sin \left(a+\frac{4}{3}\pi \right)$, with $\sin \left(3a\right)=4q/A^3$, when $4p^3+27q^2\le 0$.

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$A\cosh a$, $A\cosh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $A\cosh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\cosh \left(3a\right)=-4q/A^3$, when , , and $4p^3+27q^2>0$.

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$B\sinh a$, $B\sinh \left(a+\frac{2}{3}\pi \mathrm{i}\right)$, and $B\sinh \left(a+\frac{4}{3}\pi \mathrm{i}\right)$, with $\sinh \left(3a\right)=-4q/B^3$, when $p>0$.

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27: 12.7 Relations to Other Functions

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12.7.8 $$U(-2,z)=\frac{z^{5/2}}{4\sqrt{2\pi }}\left(2K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+3K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{5}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.11 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.9 $$U(-1,z)=\frac{z^{3/2}}{2\sqrt{2\pi }}\left(K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.10 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.11 $$U(1,z)=\frac{z^{3/2}}{\sqrt{2\pi }}\left(K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

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12.7.12 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)={\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{1}{4},\frac{1}{2},-\frac{1}{2}{z}^2),$$

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Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.2 (modification of)

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

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12.7.13 $$u_2(a,z)=z{\mathrm{e}}^{-\frac{1}{4}{z}^2}M(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=z{\mathrm{e}}^{\frac{1}{4}{z}^2}M(-\frac{1}{2}a+\frac{3}{4},\frac{3}{2},-\frac{1}{2}{z}^2).$$

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Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iv), §12.7 and Ch.12

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28: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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$\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$	$\sqrt{2}(\sqrt{3}+1)$	$\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$
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$\pi /4$	$\frac{1}{2}\sqrt{2}$	$\frac{1}{2}\sqrt{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
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$5\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$2+\sqrt{3}$	$\sqrt{2}(\sqrt{3}-1)$	$\sqrt{2}(\sqrt{3}+1)$	$2-\sqrt{3}$
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$7\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$	$\sqrt{2}(\sqrt{3}-1)$	$-\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$
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$11\pi /12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
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29: 14.4 Graphics

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Figure 14.4.1: ${𝖯}_{\nu }^0\left(x\right)$, $\nu =0,\frac{1}{2},1,2,4$. Magnify

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Figure 14.4.2: ${𝖰}_{\nu }^0\left(x\right)$, $\nu =0,\frac{1}{2},1,2,4$. Magnify

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Figure 14.4.3: ${𝖯}_{\nu }^{-1/2}\left(x\right)$, $\nu =0,\frac{1}{2},1,2,4$. Magnify

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Figure 14.4.24: ${𝑸}_0^{\mu }\left(x\right)$, $\mu =0,2,4,8$. Magnify

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Figure 14.4.28: ${𝑸}_1^{\mu }\left(x\right)$, $\mu =0,2,4,8$. Magnify

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30: 19.38 Approximations

… ►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. …